<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b3">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b4">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b3">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Olive" title="b2">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Olive" title="b1">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E2:56"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font></span><br/>

<a NAME="E3:56"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
for <font color="Olive" title="b1">r1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">r1</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/>

<div><span class="lab"><font color="Green" title="E43">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_1_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><a NAME="E2:56_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c6">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">r0</font> = <font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E4:56_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c6">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/><div><span class="lab"><font color="Green" title="E47">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_1_1_1_1_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><a NAME="E2:56_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span><br/>

<a NAME="E2:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E3:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></span>;<br/>
<a NAME="E4:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E5:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<font color="Maroon" title="c6">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E6:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="txt" href="#E5:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="extreal1.html#T39" title="EXTREAL1:th.39">EXTREAL1:39</a></span>;<br/>
<a NAME="E7:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E6:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="xxreal_3.html#T83" title="XXREAL_3:th.83">XXREAL_3:83</a></span>;<br/>
<a NAME="E8:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A19</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T82" title="XCMPLX_1:th.82">XCMPLX_1:82</a></span>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c6">r1</font>

;<br/>
<a NAME="E10:56_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:56_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E7:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E9:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="txt" href="#E10:56_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:56_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
 <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>;<br/><a NAME="E4:56_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A23</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span><br/>

<a NAME="E2:56_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A24</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E3:56_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E4:56_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="extreal1.html#T38" title="EXTREAL1:th.38">EXTREAL1:38</a></span>;<br/>
<a NAME="E5:56_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A27</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E4:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="ref" href="xxreal_3.html#T91" title="XXREAL_3:th.91">XXREAL_3:91</a></span>;<br/>
<a NAME="E6:56_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A28</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A29</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></span>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c7">r0</font>

;<br/>
<a NAME="E8:56_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A30</font></span>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E2:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a>, <a class="ref" href="mesfunc1.html#T6" target="_self" title="MESFUNC1:th.6">Th6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:56_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="txt" href="#E7:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="txt" href="#E8:56_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A30</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:56_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A31</font></span>: 
 <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>;<br/><a NAME="E6:56_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A32</font></span>: 
 <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E5:56_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">thus </span><a NAME="E7:56_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E6:56_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="ref" href="mesfunc1.html#D17" target="_self" title="MESFUNC1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><a NAME="E2:56_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A34</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A35</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span><br/>

<a NAME="E2:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A36</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def14</a></span>;<br/>
<a NAME="E3:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></span>;<br/>
<a NAME="E4:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E53">A38</font></span>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def14</a></span>;<br/>
<a NAME="E5:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E54">A39</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E4:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="extreal1.html#T39" title="EXTREAL1:th.39">EXTREAL1:39</a></span>;<br/>
<a NAME="E6:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E55">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E5:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="ref" href="xxreal_3.html#T114" title="XXREAL_3:th.114">XXREAL_3:114</a></span>;<br/>
<a NAME="E7:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E56">A41</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A42</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T82" title="XCMPLX_1:th.82">XCMPLX_1:82</a></span>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c6">r1</font>

;<br/>
<a NAME="E9:56_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A43</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:56_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="txt" href="#E6:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="txt" href="#E9:56_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:56_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A44</font></span>: 
 <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>;<br/><a NAME="E4:56_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A45</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E52">A46</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span><br/>

<a NAME="E2:56_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E53">A47</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E52">A46</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E3:56_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E54">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E52">A46</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E4:56_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E55">A49</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E54">A48</font></span></a>, <a class="ref" href="extreal1.html#T38" title="EXTREAL1:th.38">EXTREAL1:38</a></span>;<br/>
<a NAME="E5:56_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E56">A50</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E4:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E55">A49</font></span></a>, <a class="ref" href="xxreal_3.html#T116" title="XXREAL_3:th.116">XXREAL_3:116</a></span>;<br/>
<a NAME="E6:56_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E57">A51</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A52</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E57">A51</font></span></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></span>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c7">r0</font>

;<br/>
<a NAME="E8:56_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E59">A53</font></span>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E2:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E53">A47</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a>, <a class="ref" href="mesfunc1.html#T6" target="_self" title="MESFUNC1:th.6">Th6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:56_1_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E56">A50</font></span></a>, <a class="txt" href="#E7:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E58">A52</font></span></a>, <a class="txt" href="#E8:56_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:56_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A54</font></span>: 
 <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A45</font></span></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>;<br/><a NAME="E6:56_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A55</font></span>: 
 <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A44</font></span></a>, <a class="txt" href="#E5:56_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">thus </span><a NAME="E7:56_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E6:56_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="mesfunc1.html#T33" target="_self" title="MESFUNC1:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E6:56_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1_1_1_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A56</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E45">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_1_1_1_2_1_1"/>
( <font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E46">A58</font></span>: 
<font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><a NAME="E2:56_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A59</font></span>: 
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A60</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A60</font></span></a></span>;<br/>
<a NAME="E3:56_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A61</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E1:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A60</font></span></a></span>;<br/>
<a NAME="E4:56_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A62</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E49">A61</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x1</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<span class="lab"><font color="Green" title="E51">A63</font></span>: <font color="Maroon" title="c9">y</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2"><span class="lab"><font color="Green" title="E44">A56</font></span></a></span>
;<br/>
<a NAME="E7:56_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A64</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E46">A58</font></span></a>, <a class="txt" href="#E4:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="txt" href="#E6:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E51">A63</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:56_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A60</font></span></a>, <a class="txt" href="#E7:56_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E52">A64</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:56_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E48">A65</font></span>: 
( <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A59</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E4:56_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E49">A66</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A65</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">thus </span><a NAME="E5:56_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A66</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E46">A67</font></span>: 
<font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><a NAME="E2:56_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E47">A68</font></span>: 
 <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A69</font></span>: 
 <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:56_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A70</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E48">A69</font></span></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/>
<a NAME="E4:56_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A71</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E49">A70</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E5:56_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A72</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E49">A70</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></span>;<br/>
<a NAME="E6:56_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A73</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E50">A71</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E7:56_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E53">A74</font></span>: 
for <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> holds <br/>not <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_2_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> implies not <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E54">A75</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:56_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E55">A76</font></span>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E54">A75</font></span></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/>
<span class="lab"><font color="Green" title="E56">A77</font></span>: <font color="Maroon" title="c8">y</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2"><span class="lab"><font color="Green" title="E44">A56</font></span></a></span>
;<br/>
<span class="kw">thus </span><a NAME="E5:56_1_1_1_2_1_1_2_1_1"/>
not <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E46">A67</font></span></a>, <a class="txt" href="#E4:56_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E56">A77</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E8:56_1_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E51">A72</font></span></a>, <a class="txt" href="#E6:56_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="txt" href="#E7:56_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E53">A74</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:56_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A68</font></span></a>, <a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:56_1_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_1_2"><span class="lab"><font color="Green" title="E45">A57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E2:56_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" title="MESFUNC1:func.18">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E4:56"/>
<font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="mesfunc1.html#D17" target="_self" title="MESFUNC1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
